Nuclear operators on the space $C_{\mathrm{rc}}(X,E)$ of vector-valued continuous functions

Abstract

Let $X$ be a completely regular Hausdorff space and $E$ and $F$ be Banach spaces. Let $C_{\mathrm{rc}}(X,E)$ denote the Banach space of all continuous functions $f:X\to E$ such that $f(X)$ is a relatively compact set in $E$. Let ${\beta }_{\sigma }$ be the strict topology on $C_{\mathrm{rc}}(X,E)$. We characterize the nuclearity of a $({\beta }_{\sigma },\Vert \cdot {\Vert }_F)$-continuous operator $T:C_{\mathrm{rc}}(X,E)\to F$ in terms of its representing operator-valued Baire measure. As an application, we establish the relationship between the nuclearity of a $({\beta }_{\sigma },\Vert \cdot {\Vert }_F)$-continuous operator $T:C_{\mathrm{rc}}(X,E)\to F$ and the nuclearity of its conjugate operator $T^{\prime }$.